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In this letter, a new adaptive Gaussian process (GP) frame work for signal extrapolation is proposed. Signal extrapolation is an essential task in many applications such as concealment of corrupted data in image and video communications. While possessing many interesting properties, Gaussian process priors with inappropriate stationary kernels may create extremely blurred edges in concealed areas of the image. To address this problem, we propose adaptive non-stationary kernels in a Gaussian process framework. The proposed adaptive kernel functions are defined based on the hypothesized edges of the missing areas. Experimental results verify the effectiveness of the proposed method compared to the existing state of the art algorithms, based on objective and subjective evaluations.